Monotone dynamical systems

Abstract

Development is about creating something more from something less, for example, a walking and talking toddler from a helpless infant. One current theoretical framework views the developmental process as a change within a complex dynamic system. Development is seen as the emergent product of many decentralized and local interactions that occur in real time. We examine how studying the multicausality of real-time processes could be the key to understanding change over developmental time. We specifically consider recent research and theory on perseverative reaching by infants as a case study that demonstrates this approach.Contemporary developmental psychologists are still asking the same question that has intrigued philosophers and scientists since ancient times. How does the human mind, with all its power and imagination, emerge from the human infant, a creature so unformed and helpless? Some see the transformation as so remarkable that they endow infants with genetically programmed and pre-existing mental structures trapped in an immature body: latent capabilities for language, number, and physical and social reasoning that await revelation as infants mature. We also see the transformation as remarkable, but suggest that development is better understood as the emergent product of many decentralized and local interactions that occur in real time. That is, the developmental process is viewed as change within a complex dynamic system. There are several good introductions available to the concepts and mathematics of dynamic systems theory for cognitive scientists [1,2]. Development as a dynamic systemThe idea of emergence -the coming into existence of new forms through ongoing processes intrinsic to the systemare not new to developmental psychology. Developmental theorists such as Kuo, Oyama and Gottlieb have long emphasized the probabilistic, epigenetic nature of ontogenetic processes. Biologists and psychologists such as Waddington, von Bertalanffy, Lewin and Gesell have envisioned behaviour and development as morphogenetic fields that unify multiple, underlying components. But only in the past decade or so have the concepts and models of non-linear dynamic systems made in-roads into traditional developmental psychology, becoming a contender for a new developmental theory [3 -9] and fundamentally changing the way development is studied (see Box 1).Developmental psychologists have used dynamic systems ideas both as a conceptual theory [3,7,9] and in various formal mathematical treatments of developmental change. These include connectionist models [10], catastrophe theories of structural change from a neo-Piagetian perspective [11] and models based on prey -predator relationships in which skills are envisioned as arising from recursive interacting 'growers' [6,12]. Moreover, dynamic views of development have encompassed many different content domains, including mother-infant relationships, imitation, language, social relationships, perception and action, and atypical patterns of developmental change [13 -18].What u...

Distributed topology control of wireless networks

“…(2). The above lemma then is simply a restatement of Theorem 4.3, page 11, [24]. We strengthen assumption (A1) to:…”

“…(2), To see this, one mimics the proof of Theorem 1.1, page 56, of [24] with weak inequalities in place of strong, in order to verify the strong order-preserving property defined on page 2 of [24] for the flow associated with Eqn. (2).…”

Parasite sources and sinks in a patched Ross–Macdonald malaria model with human and mosquito movement: Implications for control

“…Furthermore, the crucial part in the proof in [17] (Theorem 7 on p. 47), where the global stability result of the equilibria is established, is not shown there explicitly, but instead attributed to a theorem in [21]. We note that [21] precedes the theory of monotone dynamical systems [32] which was pioneered by Morris Hirsch in the 1980-ies by almost a decade, and that [17] was published around the time monotone systems theory was being developed, but, perhaps not surprisingly, without using that theory.…”

“…Alternatively, one could exploit the monotonicity of the system, see [32], as well as the Appendix, for more on monotone systems: Theorem 1. If R 0 < 1, then all solutions of (5) − (6) converge to (0, 0).…”

“…Such system are examples of what are known as (strongly) monotone systems [32] and they have the remarkable property that their solutions satisfy a particular comparison principle stated below.…”